Periodic solutions for a class of neutral functional differential equations with infinite delay

Abstract

In this paper, we study the existence and uniqueness of periodic solutions of the nonlinear neutral functional differential equation with infinite delay of the form d d t ( x ( t ) − ∫ − ∞ 0 g ( s , x ( t + s ) ) d s ) = A ( t , x ( t ) ) x ( t ) + f ( t , x t ) . In the process we use the fundamental matrix solution of x ′ ( t ) = A ( t , u ( t ) ) x ( t ) and construct appropriate mappings, where u ∈ C ( R , R n ) is an ω -periodic function. Then we employ matrix measure and the Leray–Schauder fixed point theorem to show the existence of periodic solutions of this neutral differential equation. In the special case where g ( s , u ) ≡ 0 and A ( t , x ) = A ( t ) , some sufficient conditions which ensure the uniform stability and global attractivity of a unique periodic solution are derived.